Solve n/n-4+n=12-4n/n-4

$$\frac { n } { n - 4 } + n = \frac { 12 - 4 n } { n - 4 }$$

Solve for n

$$n=-4$$

$$n=3$$

Steps Using the Quadratic Formula

Variable n cannot be equal to 4 since division by zero is not defined. Multiply both sides of the equation by n-4.

$$n+\left(n-4\right)n=12-4n$$

Use the distributive property to multiply n-4 by n.

$$n+n^{2}-4n=12-4n$$

Combine n and -4n to get -3n.

$$-3n+n^{2}=12-4n$$

Subtract 12 from both sides.

$$-3n+n^{2}-12=-4n$$

Add 4n to both sides.

$$-3n+n^{2}-12+4n=0$$

Combine -3n and 4n to get n.

$$n+n^{2}-12=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$n^{2}+n-12=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 1 for b, and -12 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-1±\sqrt{1^{2}-4\left(-12\right)}}{2}$$

Square 1.

$$n=\frac{-1±\sqrt{1-4\left(-12\right)}}{2}$$

Multiply -4 times -12.

$$n=\frac{-1±\sqrt{1+48}}{2}$$

Add 1 to 48.

$$n=\frac{-1±\sqrt{49}}{2}$$

Take the square root of 49.

$$n=\frac{-1±7}{2}$$

Now solve the equation n=\frac{-1±7}{2} when ± is plus. Add -1 to 7.

$$n=\frac{6}{2}$$

Divide 6 by 2.

$$n=3$$

Now solve the equation n=\frac{-1±7}{2} when ± is minus. Subtract 7 from -1.

$$n=-\frac{8}{2}$$

Divide -8 by 2.

$$n=-4$$

The equation is now solved.

$$n=3 n=-4$$

Steps for Completing the Square

Variable n cannot be equal to 4 since division by zero is not defined. Multiply both sides of the equation by n-4.

$$n+\left(n-4\right)n=12-4n$$

Use the distributive property to multiply n-4 by n.

$$n+n^{2}-4n=12-4n$$

Combine n and -4n to get -3n.

$$-3n+n^{2}=12-4n$$

Add 4n to both sides.

$$-3n+n^{2}+4n=12$$

Combine -3n and 4n to get n.

$$n+n^{2}=12$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$n^{2}+n=12$$

Divide 1, the coefficient of the x term, by 2 to get \frac{1}{2}. Then add the square of \frac{1}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$n^{2}+n+\left(\frac{1}{2}\right)^{2}=12+\left(\frac{1}{2}\right)^{2}$$

Square \frac{1}{2} by squaring both the numerator and the denominator of the fraction.

$$n^{2}+n+\frac{1}{4}=12+\frac{1}{4}$$

Add 12 to \frac{1}{4}.

$$n^{2}+n+\frac{1}{4}=\frac{49}{4}$$

Factor n^{2}+n+\frac{1}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(n+\frac{1}{2}\right)^{2}=\frac{49}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(n+\frac{1}{2}\right)^{2}}=\sqrt{\frac{49}{4}}$$

Simplify.

$$n+\frac{1}{2}=\frac{7}{2} n+\frac{1}{2}=-\frac{7}{2}$$

Subtract \frac{1}{2} from both sides of the equation.

$$n=3 n=-4$$